Software used: *Stella 4d*, available at http://www.software3d.com/Stella.php.

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Software used: *Stella 4d*, available at http://www.software3d.com/Stella.php.

Software used: *Stella 4d*, available at http://www.software3d.com/Stella.php.

This cage is made of 36 identical rhombi, and has six openings. I made it with *Stella 4d*, a program available at this website.

Each of these dodecahedra were modified by truncations at exactly four of their three-valent vertices. As a result, each has four equilateral triangles as faces. In the one above, the Platonic dodecahedron’s pentagonal faces are modified into a dozen irregular hexagons by these truncations, while, in the one below, the rhombic dodecahedron’s faces are modified into twelve irregular pentagons.

Both of these polyhedra were created using *Stella 4d*, software you can try for yourself at this website.

As it turns out, eight icosahedra form this rhombic ring, by augmentation:

Measured from the centers of these icosahedra, the long and short diagonal of this rhombus are in a (√2):1 ratio. How do I know this? Because that’s the only rhombus which can made this polyhedron, a rhombic dodecahedron, dual to the cuboctahedron.

This rhombic dodecahedral cluster of icosahedra could be extended to fill space, since the rhombic dodecahedron itself has this property, an unusual property for polyhedra. Whether space-filling or not, the number of icosahedron per rhombic-dodecahedron edge could be increased to 5, 7, 9, or any greater odd number. Why would even numbers not work? This is a consequence of the fact that opposite faces of an icosahedron are inverted, relative to each other; a pair of icosahedra (or more than one pair, producing odd numbers > 1 when added to the vertex-icosahedron) must be attached to the one at a rhombic-dodecahedron-vertex to make these two inversions bring the triangular face back around to its original orientation, via an even number of half-rotations, without which this consruction of these icosahedral rhombi cannot happen.

Here’s another view of this rhombic dodecahedron, in “rainbow color” mode:

All images above were produced using *Stella 4d*, software which may be tried for free right here.

The dual to this cluster-polyhedron appears below. Both virtual models were created using *Stella 4d: Polyhedron Navigator*, software available here.

*Stella 4d* has this in its built-in library. You can try it at http://www.software3d.com/Stella.php.

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